Question

What is the least common multiple of 16 and 24?

Answer 1

The lowest common multiple (LCM) of `16` and `24` is `48`.

Explanation:

The prime factorization of `16`:
`16 = 2*2*2*2 = 2^4`

The prime factorization of `24`:
`24 = 2 * 2 * 2 *3 = 2^3*3`

Both `16` and `24` have `2^3` (`8`) in common, so we can remove it from one of the numbers, leaving `2^4` and `3`. There are no more numbers in common, so the lowest common multiple is `2^4 * 3 = 48`.

Answer 2

A very different way of thinking about it!

48

Explanation:

Think of 16 as `16/24->8/12->4/6->2/3` of 24

`2/3->` Not a whole number

`2/3+2/3->4/3` Not a whole number

`2/3+2/3+2/3=6/3->` whole number of 2

`2xx24=48`
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`color(blue)("Why does this work?")`

Although it is not obvious the above process is linked to what follows.

16 is a portion of 24 in that `16+8=24`

So we cycle through occurrences of `16+8` until we have values such that `16` will divide exactly into the sum of the 8's. The 8's being part of the 24 means we are counting the 24's.

`16+8 = 24`
`ul(16+8=24)`
`32+16=48`

Within the 48 is another complete 16. In that we now have 3 lots of 16.